Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators

In this paper we construct the fundamental solution for a class of subelliptic operators by means of Levi’s parametrix method. We then use the fundamental solution, together with a mean value formula, to prove a maximum principle and to establish an Harnack inequality for divergence form subelliptic operators. To prove these results, we rely on some properties of the sub-Riemannian geometry relevant to the subelliptic operators.